Thick smoke from a kitchen fire killed five members of a family yesterday after it blocked the only escape route from their Chelsea apartment.

The only survivor – a 10-year-old boy – was fighting for his life at Jacobi Medical Center after being resuscitated by firefighters who smashed their way into the seventh-floor home in the Fulton Houses, on West 18th Street, at 6:30 a.m.

Rescuers found 34-year-old Delkis Balbuena lying in a bathtub full of water, her 8-year-old daughter, Nanny, lying beneath her.

Under the bathroom sink, they discovered the body of 3-year-old Bet-El. The girl had apparently crawled into the space in a desperate effort to escape the choking fumes.

Breaking into the room next door, the firefighters found the body of father Maschay Valdez, 40, slumped at the base of the kids’ bunk beds next to a window.

Beside him lay 15-month-old Ruth and the lone survivor, Jonzan, 10, who was barely hanging on.

A grief-stricken relative said Delkis was a devoted mother whose children enjoyed playing outdoors.

“Their mother loved them very much,” said Miguel Saeteros, Delkis’ brother-in-law. “She was very dedicated to those kids. They were four beautiful kids.”

The horrific discovery devastated even hardened FDNY veterans.

“The firefighters coming down had a look on their faces like they were overwhelmed,” said neighbor Miguel Acevedo, 48. “The last one came out and got on his knees and started praying.

Deputy Fire Chief James Daly described the hell into which his brave men ventured to find the family.

“The smoke was so acrid it was burning my eyes,” he said. “It was ink black and high heat.

“[The family] definitely woke up and tried to make their way out of the apartment, but the door was blocked by fire.”

The victims suffered no burns but the toxic fumes gave them little chance of survival, the FDNY said.

Firefighters broke into the home only four minutes after a 911 call was received, but were already too late.

Daly theorized that instead of closing the bedroom door, sealing gaps under it and waiting for rescuers – actions the FDNY would recommend – the family tried to get out and was beaten back by the smoke.

“There were no fire escapes,” he said. “The fire was rolling across the ceiling.”

The apartment is a “fireproof building,” meaning it’s been built with noncombustible materials and designed to stop flames spreading to neighboring units.

But that didn’t stop the blaze from engulfing the apartment.

First responders found a smoke detector melted by the furnace and hanging off the ceiling, in the passage outside the bathroom. It was supposed to be wired into the building’s power source, but had been disconnected.

“Obviously that played a big part,” Daly said.

A spokeswoman for the city Housing Authority, the building’s landlord, said the detector was replaced in February. It was working when checked by the authority two months later, she said.

An FDNY source said investigators are looking at candles and incense lit by the family as a possible source of the blaze.